Howe finiteness conjecture for covering groups
Let $G$ be the group of rational points of a connected reductive $p$-adic group. Suppose $K\subset G$ is a compact subgroup, and $\Omega\subset G$ is a closed subset that is compact modulo conjugation by $G$, in the sense that $\Omega = \{gxg^{-1} \mid g\in G, x\in\omega\}$ for some compact subset $\omega \subset G$. Let $J(\Omega)$ denote the space of $G$-invariant distributions on $G$ that are supported on $\Omega$, and $\mathcal{H}(K \backslash G/ K)$ the algebra of compactly supported, $K$ bi-invariant, complex-valued functions on $G$. A fundamental statement in harmonic analysis on $p$-adic groups is that the restriction of $J(\Omega)$ to $\mathcal{H}(K\backslash G/ K)$ is finite dimensional. This statement was conjectured by Howe, proved by Clozel, and later proved again by Barbasch-Moy via different means. The paper under review proves the Howe conjecture in the case where we replace the group $G$ by a finite central extension $\widetilde{G}$ of $G$ by $\mu_n$. It is enough to prove an equivalent statement about regular semisimple orbital integrals, since these are known to be dense in the space of invariant distributions. In order to adapt the method of Clozel, it is necessary to prove covering-group versions of the $K$-finiteness of elliptic representations; Clozel's integration formula (which expresses a distribution character of a representation in terms of those of its Jacquet modules); Casselman's character formula (which does the same for characters expressed as functions on the group); and some easier results as well.
scientific article; zbMATH DE number 6900923
scientific article